Note: This is about a front-loading, rather new machine (purchased in 2013, according to my landlady)
Most of the functions should be rather transferable for other front-loading machines, but if you have any questions about your particular machine, you can feel free to email me a picture of your particular machine.

I'll start on right and move to the left.
The most important button is the one on the far right, the 전원 (power) button. Without turning the machine on, you can't use any of the other buttons.

The small button in the upper right corner, 예약, is useful to schedule a wash load.
You can push this button and select when you'd like to schedule the next load (ie. 60 minutes later, 30 minutes later, etc).

The 물온도 selects the water temperature of your wash. Depending on the type of wash you select (with the dial on the left), some temperatures are not options, you'll have to see on your particular machine. The bottom choice is 냉수 or cold water.

The 탈수 button selects the strength of your spin cycle itself. Since most Korean houses do not have dryers, this can be useful, but it can also kind of take a toll on your fabrics. On my machine, it goes from strongest (강) at the top to none at the bottom.

The play/pause button does just that.

I don't usually mess with the buttons above it, but they do come in handy sometimes.
온수헹굼: Hot water rinse
예비세탁: Prewash
강력: Power wash
구김방지: Wrinkle-release

Ok, the most important...that crazy dial with so many different options (clockwise from the top)
표존: Standard wash

I have noticed that many of the wash cycles are quite long, maybe around 2 hours for some, but the standard setting usually takes about 1 hour 15 minutes for me.

After you choose your specific setting you would like and push start, the machine typically weighs your laundry and determines how long your wash will take.

If you have somehow chosen the wrong setting, you can usually pause the wash partway through and change it (from experience).

Korean houses/dorms/sharehouses do not usually have dryers.
So, you'll probably have to buy a drying rack for your laundry. Most dorms do provide them for you, but they are probably around 10,000KRW (for a small one) or 14,000KRW for a medium one like I have. There are also crazy racks that are like three tiers or something, they'll run you about 30,000KRW.

Korean households typically dry their laundry on their verandas, but if you don't have a veranda, another option is turning on your ondol (in-floor heating) to speed up the process.

I also highly recommend using fabric softener. I never used it in the US, but I realized I needed it after moving here, if for nothing else than making sure your clothes smell fresh.

Oh, and I had a request to talk about laundry detergent and stuff here, too.

The Korean word is 세탁세제 (say-tak say-jay)
I usually use liquid detergent, just a personal preference.
I recommend the brand, LiQ, because I have found it to be less harsh on my clothes than many of the other brands.

I really do not like the 비트 brand, it really damaged my dark clothing!

There are plenty of Korean brands for fabric softener (섬유 유연제 - sum-you you-yeon-jay), but I like Downy's scents, so I stick with that brand. It's a little more expensive than something like Pigeon (피죤) or 쉐리.

Oh, for wool/handwash items, you'll need to buy some special shampoo, the word is 울세제 - ool say-jay.
There are not that many brands here. I usually use just the 울샴푸 brand ("wool shampoo").

I think that's about it. Let me know if you have any other questions!
Oh, by the way, I typically buy my laundry supplies online. You'll get a much better price and not have to carry it from the store!